All copyrightable works published in the United States before 1923 are in the public domain; works created before 1978 but not published until recently may be protected until 2047. For works that received their copyright before 1978, a renewal had to be filed in the work's 28th year with the Library of Congress Copyright Office for its term of protection to be extended.
The need for renewal was eliminated by the Copyright Renewal Act of 1992, but works that had already entered the public domain by non-renewal did not regain copyright protection. Therefore, works published before 1964 that were not renewed are in the public domain. With rare exception (such as very old works first published after 2002), no additional copyrights will expire (thus entering the public domain) until at least 2019 due to changes in the applicable laws.
Many local societies and individual genealogists are now in or thinking of being in the scanning business to get local records online. This copyright question comes up sooner or later.
The above statement is about all you need to know. If you scan publications in print before 1923, you are safe. The question comes with materials that are unmarked or marked after 1923.
There is a caveat in the law that refers to using parts of material for references or reviews, as in footnotes or book reviews. That can get hazy--they never say how much is too much. You're on your own there.
Another problem to me are small publications that are common in genealogy, like an old marriage list or cemetery list, for instance, where the author has since died. Did they pass the copyright on to heirs or did they not? And how do you find them? And, if you scan some of those pubs, which probably did not make anyone a dime anyway, will someone come after you with a cease and desist order?
There are tons of small genealogy publications sitting out there that would make great items to place on local society websites, many of them printed in the 1950's and '60's and '70's. All of them in this Catch 22 position. Do you just scan them, keep your fingers crossed, and wait for the time when you have to say, "Oops," and apologize?
Could be that now we can be glad that lawyers cost so much. Will anyone hire a pricy shyster to get blood out of a turnip? I'm familiar with many local society treasuries, and you're definitely talking turnips! So, maybe you should just go for it--maybe the gain in data is worth the risk. And that's all I have to say about that...